Solid wall simulations for dam-break floods

The initial validity of the MPM results for dam-break flows was established by direct compari- son with results produced using SPH published in Dongfang Liang’s 2009 paper “Evaluating shallow water assumptions in dam-break flows”[88], and with published results in Dongfang

4.3 Model Validation 101

Fig. 4.12 Initial geometries of solid wall verification simulations replicating Liang (2009) [88] and Cruchaga et al. (2006) [33]

Liang and Xuanyu Zhao’s 2017 paper “Numerical simulations of dam-break floods with MPM” [89].

The results produced are also compared to numerical and experimental results published in Cruchaga, Celentano and Tezduyar’s 2006 paper “Collapse of a liquid column: numerical simulation and experimental validation”[33], experimental results published in Martin and Moyce (1952) “An experimental study of the collapse of liquid columns on a rigid horizontal plane”[104] and numerical results published in Koshizuka and Oka (1996) “Moving-Particle Semi-Implicit Method for Fragmentation of Incompressible Fluid”[75].

These simulations use a solid end boundary and form the basis of the simulations in section

4.3.2, on porous wall simulations. Figure 4.12 shows the initial geometry used for these validation simulations, based on the geometries used in Liang (2009) [88] and in Cruchaga et al. (2006) [33].

Figure 4.13 shows a comparison of particle distribution at different time intervals for these simulations (right-hand side) with those published by Liang (left-hand side). In both the SPH and MPM simulations, the initial pressure distribution is hydrostatic: the pressure contours are plotted for the MPM results in the first instance and the hydrostatic pressure distribution is demonstrated by the horizontal pressure contours that run parallel to the free- surface. Figures4.13(b)-(e) show the evolution of the flood front as it propagates down the tank. The shape of the flood front produced by the MPM results is in very good agreement with those produced using the SPH model. Once the flood hits the end of the tank, it rapidly climbs the wall, reaching much higher than the initial height of the column, shown in Figure4.13(f), as

the kinetic energy of the front is transformed into potential energy. Again, both the MPM and SPH models are in good agreement with the mechanism by which the flood evolves, however, the lack of smoothing in the MPM simulations means that particle scattering can be observed. The next phase is for the run-up water to collapse down and curl back on itself, shown in Figure4.18(g), and finally, this body of water plunges into the wave generated by the initial column collapse, that is still moving towards the end of the tank, forming a cavity and forceful splashing, resulting in turbulent motion. Both models show the same progression, although once again particle scattering is evident in the MPM simulations. Overall, good agreement is shown with the particles following the same distribution, however, the MPM results exhibit significantly more particle scattering after the impact with the solid wall. The pressure contours are much smoother in the SPH simulation, whereas in the MPM simulations the pressure fluctuations are too high to be useful. However, when we compare the shape of the plume and the run-up height, the SPH and MPM methods show good agreement. As noted by Liang, due to the absence of the vertical acceleration, the SWEs solver cannot account for the time required to transform the kinetic energy of the wave into potential energy, for which the vertical movement is essential [88]. Consequently, the SWEs solver unsurprisingly gives very different results to the other two models, especially after the impact with the solid wall.

Figure4.14shows a quantitative comparison of the propagation speed of the flood front over time with results published by various research groups using both experimental and numerical methods. These results are normalised according to the Froude scaling law to allow for comparison with results from different geometries so that

t∗= _qt

H1

g

, x∗= x

H₁ (4.4)

where H1 is the initial height of the water column. Since the SWEs results have been

demonstrated to be unrealistic, these are not plotted here. As noted by Liang, the SPH results depend slightly on the kernel smoothing length, with a smaller smoothing length producing a higher speed [88]. The MPM results show the same trend with a slightly higher speed, this may be explained by a small amount of particle scattering at the leading edge making it challenging to identify the precise location of the flood front. Generally, the numerical simulations are in very good agreement with experimental results, and with results produced using other numerical methods.

Figures4.15and4.16show a comparison for the time history of the flood front and column height respectively for the MPM simulations using the geometry after the experiments published in Cruchaga et al. [33], shown in figure4.12. Koshizuka and Oka use an aspect ratio α = 2 and a tank length four times the width of the column for a total of 14.6cm [75]. Again, the

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Fig. 4.13 Direct comparison of results produced using MPM with the extensively verified SPH results produced by Liang (2009) [88]. In the left column, solid lines with circles represent the shallow water equations (SWEs) results and the contours represent SPH results. The MPM results are plotted in the right columns: (a) t = 0s; (b) t = 0.03s; (c) t = 0.07s; (d) t = 0.10s; (e) t = 0.15s; (f) t = 0.35s; (g) t = 0.55s; (h) t = 0.66s

Fig. 4.14 Comparison of results from simulations using the geometry from Liang (2009) to published experiments, plotting the evolution of the position of the front over time

Fig. 4.15 Comparison of results from simulations using the geometry from Cruchaga et al. (2006) to published experiments, plotting the evolution of the position of the front over time

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Fig. 4.16 Comparison of results from simulations using the geometry from Cruchaga et al. (2006) to published experiments, plotting the evolution of the height of the column over time

results have all been normalised according to the Froude scaling law to allow for precise comparison. The MPM results are a very close match for the numerical results published by Koshizuka and Oka (1996) [75], although the experimental results published by Martin and Moyce (1952) [104] and the experimental results published by Cruchaga et al. (2006) [33] show a slightly higher frontal propagation speed. The numerical results published by Cruchaga et al. (2006) show a noticeably initial slower propagation speed than those produced by their experiments [33]. They suggest that this occurs at least in part due to the gate opening effect; whereby the physical gate present in the experiments takes a finite amount of time to rise, affecting the initial progress of the column collapse when compared to the instantaneous gate opening of the numerical simulations. When a gate-opening parameter was included in their simulations, more advanced flood positions at early instants of the analysis are obtained at the bottom of the tank, due to the “orifice effect” induced at the beginning of the gate opening, which increases the fluid velocity at the bottom of the tank [33]. This situation quickly changes as the dam-break wave propagates and frictional effects slow the overall propagation, causing the wave to reach the end of the tank later than if no gate-opening effect is included. This would suggest that the slightly higher speed reached by the experimental results compared with the numerical results is an effect of the gate-opening in an experiment being non-instantaneous.

Investigating the evolution of the height of the water column (Figure4.16) shows a very close match between the MPM simulations and the experimental results published in Martin and Moyce (1952), suggesting that whilst the gate opening being non-instantaneous may affect the bottom of the water column as the flow develops, there is little effect on the top of the column.